Chapter 12, Weak Interactions Video Solutions, Quarks And Leptons. An Introductory Course In Modern Particle Physics

Problem 1

Give the $\pi^{+}$and $\mu^{+}$decay processes. List the possible decay modes of the $\tau^{-}$lepton (the $\tau$ is the third lepton in the sequence $e, \mu, \tau$ with a mass $m_{\tau}=1.8 \mathrm{GeV}$ ).

Narayan Hari

Numerade Educator

03:09

Problem 2

Show that a (charge-lowering) weak current of the form
$$
\bar{u}_{e} \gamma^{\mu \frac{1}{2}}\left(1-\gamma^{5}\right) u_{\nu}
$$
involves only left-handed electrons (or right-handed positrons). In the relativistic limit $(v \approx c)$, show that the electrons have negative helicity.

David Walther

Numerade Educator

09:37

Problem 3

Show that the charge-raising weak current
$$
J^{\mu}=\bar{u}_{\nu} \gamma^{\mu} \frac{1}{2}\left(1-\gamma^{5}\right) u_{e}
$$
couples an ingoing negative helicity electron to an outgoing negative helicity neutrino. Neglect the mass of the electron. Besides the configuration $\left(\mathrm{e}_{L}^{-}, \nu_{L}\right)$, show that $J^{\mu}$ also couples the following (ingoing, outgoing) lepton pair configurations: $\left(\bar{\nu}_{R}, \mathrm{e}_{R}^{+}\right),\left(0, \nu_{L} \mathrm{e}_{R}^{+}\right)$, and $\left(\mathrm{e}_{L}^{-} \bar{\nu}_{R}, 0\right)$.

Further, show that the charge-lowering weak current, (12.9), is the hermitian conjugate of (12.12):
$$
J_{\mu}^{\dagger}=\bar{u}_{e} \gamma_{\mu} \frac{1}{2}\left(1-\gamma^{5}\right) u_{\nu} .
$$
List the lepton pair configurations coupled by $J_{\mu}^{\dagger}$.
Weak interaction amplitudes are of the form
$$
\mathscr{K}=\frac{4 G}{\sqrt{2}} J^{\mu} J_{\mu}^{\dagger}
$$
Charge conservation requires that $9 \mathbb{R}$ is the product of a charge-raising and a charge-lowering current; see, for example, (12.10) and (12.11). The factor 4 arises because the currents, (12.13), are defined with the normalized projection operator $\frac{1}{2}\left(1-\gamma^{5}\right)$ rather than the old-fashioned $\left(1-\gamma^{5}\right)$.

Robert Zaballa

Numerade Educator

02:11

Problem 4

Verify the isospin factor $\sqrt{2}$ in (12.19). Note that ${ }^{14} \mathrm{C}$, ${ }^{14} \mathrm{~N}^{*},{ }^{14} \mathrm{O}$ form an isospin triplet, which can be viewed as $\mathrm{nn}, \mathrm{np}, \mathrm{pp}$, together with an isospin zero ${ }^{12} \mathrm{C}$ core (see Fig. 2.2). Keep in mind that for indistinguishable proton decays, we must add amplitudes, not probabilities.

Suzanne W.

Numerade Educator

07:41

Problem 5

Calculate $G$ from the data for the $\beta$-transition ${ }^{10} \mathrm{C} \rightarrow$ ${ }^{10} \mathrm{~B} * \mathrm{e}^{+} \nu$. The measured half-life is $\tau \log 2=20$ sec, and $E_{0}=2 \mathrm{MeV}$. $\left({ }^{10} \mathrm{C}\right.$ and ${ }^{10} \mathrm{~B}^{*}$ are both isospin $1, J^{P}=0^{+}$states. $)$

Mayank Tripathi

Numerade Educator

04:13

Problem 6

Accepting the vector boson exchange picture of weak interactions with coupling, $g=e$, estimate the mass $M_{W}$ of the weak boson. (In the standard model of weak interactions, introduced in Chapter 13, $g \sin \theta_{W}=e$, with $\left.\sin ^{2} \theta_{W} \approx \frac{1}{4} .\right)$

Deepak Kohli

Numerade Educator

02:04

Problem 7

Verify these results.

Manik Pulyani

Numerade Educator

14:04

Problem 8

Derive (12.34) by performing the $d \omega$ integration on the right-hand side (see Exercise 6.7).
Using (12.31) and (12.29), we find the spin-averaged probability is
$$
\overline{|\mathscr{M}|^{2}} \equiv \frac{1}{2} \sum_{\text {spins }}|\mathscr{R}|^{2}=64 G^{2}\left(k \cdot p^{\prime}\right)\left(k^{\prime} \cdot p\right),
$$
where $p=p^{\prime}+k+k^{\prime}$ on account of the $d^{4} k$ integration performed in (12.33). Since $m_{\mu}>200 m_{e}$, we can safely neglect the mass of the electron.

Mahnoor Amin

Numerade Educator

05:54

Problem 9

Verify (12.35). Neglect the mass of the electron, but not that of the muon.

Sai Chaitanya Tadepalli

Numerade Educator

02:02

Problem 10

Show that
$$
2\left(k \cdot p^{\prime}\right)\left(k^{\prime} \cdot p\right)=\left(p-k^{\prime}\right)^{2}\left(k^{\prime} \cdot p\right)=\left(m^{2}-2 m \omega^{\prime}\right) m \omega^{\prime}
$$
in the muon rest frame, where $p=(m, 0,0,0)$.
Gathering these results together, the decay rate in the muon rest frame is
$$
\begin{aligned}
d \Gamma=& \frac{G^{2}}{2 m \pi^{5}} \frac{d^{3} p^{\prime}}{2 E^{\prime}} \frac{d^{3} k^{\prime}}{2 \omega^{\prime}} m \omega^{\prime}\left(m^{2}-2 m \omega^{\prime}\right) \\
& \times \delta\left(m^{2}-2 m E^{\prime}-2 m \omega^{\prime}+2 E^{\prime} \omega^{\prime}(1-\cos \theta)\right)
\end{aligned}
$$
and, as for $\beta$-decay, we can replace $d^{3} p^{\prime} d^{3} k^{\prime}$ by
$$
4 \pi E^{\prime 2} d E^{\prime} 2 \pi \omega^{\prime 2} d \omega^{\prime} d \cos \theta
$$

Narayan Hari

Numerade Educator

03:21

Problem 11

Draw a diagram showing the particle helicities in the $\mu^{-}$rest frame in the case where the emitted electron has its maximum permissible energy. In this limit, explain why the electron angular distribution has the form $1-P \cos \alpha$, where $\mathbf{P}$ is the polarization of the muon and $\alpha$ is the angle between the polarization direction and the direction of the emitted electron:
$$
P \equiv \frac{N_{+}-N_{-}}{N_{+}+N_{-}}
$$
where $N_{\pm}$are the numbers of spin-up, spin-down muons.

Linda Winkler

Numerade Educator

01:59

Problem 12

"Predict" the rate for the decay $\tau^{-} \rightarrow \mathrm{e}^{-} \bar{\nu}_{e} \nu_{\tau}$, where the $\tau$-lepton has mass $1.8 \mathrm{GeV}$. The observed branching ratio of this decay mode is approximately $20 \%$. Calculate the lifetime of the $\tau$-lepton. Can you explain this branching ratio?

Carlos Henrique De Lima

Numerade Educator

04:00

Problem 13

Predict the ratio of the $\mathrm{K}^{-} \rightarrow \mathrm{e}^{-} \bar{\nu}_{e}$ and $\mathrm{K}^{-} \rightarrow \mu^{-} \bar{\nu}_{\mu}$ decay rates. Given that the lifetime of the $\mathrm{K}^{-}$is $\tau=1.2 \times 10^{-8} \mathrm{sec}$ and the $\mathrm{K} \rightarrow \mu \nu$ branching ratio is $64 \%$, estimate the decay constant $f_{K^{.}}$. Comment on your assumptions and on your result.

Sinisa Stura

Numerade Educator

00:47

Problem 14

On purely dimensional grounds, show that the cross section (for a point interaction) must behave as $\sigma\left(\nu_{e} \mathrm{e}^{-}\right)-G^{2} s$ at high energies. Comment on the significance of this result.

David Collins

Numerade Educator

04:03

Problem 15

Show that
$$
\sigma\left(\nu_{e} \mathrm{e}^{-}\right) \approx\left(E_{p} \text { in } \mathrm{GeV}\right) \times 10^{-41} \mathrm{~cm}^{2}
$$
where $E_{y}$ is the laboratory energy of the neutrino.

Ren Jie Tuieng

Numerade Educator

03:31

Problem 16

If the weak current had had a $V+A$ structure, $\gamma^{\mu}(1+$ $\left.\gamma^{5}\right)$, show that
$$
\frac{d \sigma}{d \Omega}=\frac{G^{2} s}{4 \pi^{2}}(1+\cos \theta)^{2}
$$Fig. $12.10$ Backward scattering in the center-of-mass frame. The long arrows represent the particle momenta and the short arrows represent their helicities in the limit in which the masses are negligible. The $z$ axis is along the incident neutrino direction.

Vipender Rao

Numerade Educator

06:05

Problem 17

Using the above approach, show that
$$
\Gamma\left(\pi^{-} \rightarrow \pi^{0} \mathrm{e}^{-} \bar{\nu}_{e}\right)=\frac{G^{2}}{30 \pi^{3}}(\Delta m)^{5},
$$
where $\Delta m=m\left(\pi^{-}\right)-m\left(\pi^{0}\right)=4.6 \mathrm{MeV}$. Evaluate the decay rate and compare with $\Gamma\left(\pi^{-} \rightarrow \mu^{-} \bar{\nu}_{\mu}\right)$.
Fig. $12.12$ The quark diagram responsible for neutron $\beta-$ decay, $\mathrm{n} \rightarrow \mathrm{pe}^{-} \bar{v}_{e}$. The two spectator quarks which do not take part in the weak interaction are shown by dashed lines.

Declan Nell

Numerade Educator

02:21

Problem 18

Show that deep inelastic electron electromagnetic scattering on an isoscalar target gives
$$
\frac{d \sigma(\mathrm{eN} \rightarrow \mathrm{eX})}{d x d y}=\frac{2 \pi \alpha^{2}}{Q^{4}} x s\left[1+(1-y)^{2}\right] \frac{5}{18}[Q(x)+\bar{Q}(x)]
$$
per nucleon, see Exercise 9.5. Note that, in contrast to $\nu \mathrm{N} \rightarrow \mu \mathrm{X}$, (12.78) embodies parity conservation so $Q$ and $\bar{Q}$ appear symmetrically.

Ashwin Banarsee

Numerade Educator

07:05

Problem 19

If $\sigma(\bar{\nu}) / \sigma(\nu)=R$, show that
$$
\frac{\int x \bar{Q}(x) d x}{\int x Q(x) d x}=\frac{3 R-1}{3-R}
$$
Detailed analyses show that the functions $u(x), d(x), \ldots$, are indeed the same whether one extracts them from electroproduction or neutrino experiments. This is a definitive success of the parton model: the $u(x), d(x)$, describe the intrinsic structure of the hadronic target and are the same whatever experimental probe is used to determine them.

Rashmi Sinha

Numerade Educator

02:30

Problem 20

Show that $|T R(\nu \mathrm{q} \rightarrow \nu \mathrm{q})|^{2}$ behaves like $s^{2}, s^{2}(1-y)^{2}$, $s^{2} y^{2}$ for pure $V-A$, pure $V+A$, and $S, P$ neutral couplings of the quark, respectively. Pure $V \pm A$ denote $\gamma^{\mu}\left(1 \pm \gamma^{5}\right)$ couplings, and $S, P$ stands for the scalar, pseudoscalar interaction amplitude
$$
\text { II }=\frac{G_{N}}{\sqrt{2}}\left(\bar{u}_{\nu}\left(1-\gamma^{5}\right) u_{\nu}\right)\left(\bar{u}_{q}\left(g_{s}-g_{P} \gamma^{5}\right) u_{q}\right) .
$$

Guilherme Barros

Numerade Educator

02:50

Problem 21

Estimate the relative rates for the following three decay modes of the $\mathrm{D}^{0}(\mathrm{cu})$ meson: $\mathrm{D}^{0} \rightarrow \mathrm{K}^{-} \pi^{+}, \pi^{-} \pi^{+}, \mathrm{K}^{+} \pi^{-}$.

Sarah Mccrumb

Numerade Educator

01:24

Problem 22

Given that the partial rate
$$
\Gamma\left(\mathrm{K}^{+} \rightarrow \pi^{0} \mathrm{e}^{+} \nu\right)=4 \times 10^{6} \mathrm{sec}^{-1},
$$
calculate the rate for $\mathrm{D}^{0} \rightarrow \mathrm{K}^{-} \mathrm{e}^{+} \nu$. Hence, estimate the lifetime of the $\mathrm{D}^{0}$ meson.

Kai Chen

Princeton University

03:22

Problem 23

Show, in the "spectator" quark model approach, that the charmed meson lifetimes satisfy
$$
\tau\left(D^{0}\right)=\tau\left(D^{+}\right)=\tau\left(F^{+}\right),
$$
where $\mathrm{F}^{+}$is made of $\mathrm{c}$ and $\overline{\mathrm{s}}$ quarks, see Chapter 2 .

Suzanne W.

Numerade Educator

View

Problem 24

Verify (12.125) using (5.39).

Victor Salazar

Numerade Educator

02:48

Problem 25

Show that $C=-1$ for a photon and hence that $C=$ $+1$ for $a \pi^{0}$.

Hint Under $e \rightarrow-e$, the amplitude corresponding to Fig. 1.9a will change sign.

Nicole Smina

Numerade Educator

04:20

Problem 26

Show that in the absence of angular momentum, a $\pi^{+} \pi^{-}$or $\pi^{0} \pi^{0}$ state is an eigenstate of $C P$ with eigenvalue $+1$. Further, show that by adding an $\mathrm{S}$ wave $\pi^{0}$, we obtain $C P$ eigenstates $\pi^{+} \pi^{-} \pi^{0}$ or $3 \pi^{0}$ with eigenvalue $-1$.

Hafiz Shahzaib

Numerade Educator